LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-08-11

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The positive and negative variations are nondecreasing and give the Jordan identities

Statement

For a bounded-variation function $f : [a, b] \to R$ , the functions $P_{f}$ and $N_{f}$ are nondecreasing and

$f (x) = f (a) + P_{f} (x) - N_{f} (x), V_{f} (x) = P_{f} (x) + N_{f} (x) .$

Both $P_{f} (a)$ and $N_{f} (a)$ are $0$ .

Facts & Assumptions

Given: A bounded-variation function $f : [a, b] \to R$ and its functions $V_{f}, P_{f}, N_{f}$ .

[L1]

$V_{f}, P_{f}, N_{f}$ are defined by the displayed formulas in Variation function and positive and negative variations.

[L2]

For $x \leq y$ , $V_{f} (y) - V_{f} (x) = {Var}_{[x, y]} (f)$ (Total variation is additive over adjacent subintervals and decreases under restriction).

[L3]

$∣ f (y) - f (x) ∣ \leq {Var}_{[x, y]} (f)$ (Total variation bounds increments; bounded-variation functions are bounded; zero variation means constant).

[L4]

A function is nondecreasing when $x \leq y$ implies $g (x) \leq g (y)$ (Nondecreasing, increasing (strictly increasing), nonincreasing, decreasing, monotone and strictly monotone real functions on a subset of $R$ , with the dictionary to monotone sequences).

[L5]

$∣ u ∣ \geq u$ and $∣ u ∣ \geq - u$ (Absolute value in an ordered field).

Proof

technique · direct

1.1L1L2L3L4L5

For $x \leq y$ , [L2] and [L3] give $V_{f} (y) - V_{f} (x) \geq ∣ f (y) - f (x) ∣$ , hence this difference is at least both $f (y) - f (x)$ and $f (x) - f (y)$ . Therefore $P_{f} (y) - P_{f} (x) \geq 0$ and $N_{f} (y) - N_{f} (x) \geq 0$ , so both functions are nondecreasing.

1.2L1algebra

Adding and subtracting the defining formulas gives $P_{f} (x) + N_{f} (x) = V_{f} (x)$ and $P_{f} (x) - N_{f} (x) = f (x) - f (a)$ . At $x = a$ , $V_{f} (a) = 0$ , so $P_{f} (a) = N_{f} (a) = 0$ .

2.1step 1.2algebra∎

Rearranging the second identity in step 1.2 gives $f = f (a) + P_{f} - N_{f}$ , while the first is the asserted variation identity.

Depends on

Dependency tree · 2 levels

Sources

Christopher Heil, Absolute Continuity and the Banach-Zaretsky Theorem